有理顶点算子代数的扩张及扭模理论

本课题研究有理顶点算子代数的表示理论及相关议题，包括扩张问题及扭模理论。主要内容有三个方面：1）研究一类由顶点算子代数所确定的fusion张量范畴的极小模扩张，确定极小模扩张的存在性和分类；2）研究有理顶点算子代数的扩张。拟证明一个有理顶点算子代数只有有限个扩张并利用模张量范畴中的代数分类扩张；3）研究顶点算子代数的扭模理论。首先对顶点算子代数的任何一个有限阶自同构构造扭模，并将扭模的概念推广到超群，从而研究一个有理顶点算子代数的任何子代数的模范畴。本项目所研究的问题是顶点算子代数的热点问题，研究和解决这些问题对顶点算子代数理论的发展相当重要，并为物理中的二维共形场论奠定了坚实的数学基础。

This is a proposal on the representation theory of rational vertex operator algebra and related topics, including the extensions and twisted module theory. The proposal consists of three major parts: 1）Study minimal modular extensions of certain fusion tensor category determined by a rational vertex operator algebra. Prove the existence of minimal modular extensions and classify the minimal modular extensions. 2）Study the extensions of a rational vertex operator algebra. Prove that there are only finitely extensions and using the modular tensor category to classify the extensions. 3）Establish general twisted module theory. Construct twisted module for any finite order automorphism of a vertex operator algebra, extend the concept of twisted modules from group to hypergroup, and investigate the module category of a subalgebra of arbitrary rational vertex operator algebra. This proposal will deal with various hot topics in the field. Studying and solving these problems successfully will not only important for the development of the theory of vertex operator algebras, but also provide solid mathematical foundation to the 2-dimensional conformal field theory.